This paper has three aims. First, for \(n \ge 1\) we construct a family of real-rooted trigonometric polynomial maps \(P: \mathbb {C}^n \mapsto \mathbb {C}^n\) whose divisors are Fourier Quasicrystals (FQ). For \(n = 1\) these divisors include the first nontrivial FQ with positive integer coefficients constructed by Kurasov and Sarnak (J Math Phys 61:083501, 2020), and for \(n > 1\) they overlap with Meyer’s curved model sets (Meyer, in: Flandrin et al., Theoretical physics, wavelets, analysis, genomics. applied and numerical harmonic analysis, Birkhäuser, Cham, 2023) and two-dimensional (Meyer, Publ Mat 67(1):469–480, 2023) and multidimensional (Meyer, Trans R Norw Soc Sci Lett 1:1–24, 2023) crystalline measures. We prove that the divisors are FQ by directly computing their Fourier transforms using a formula derived in Lawton (J Geom Anal 32:60, 2022). Second, we extend the relationship between real-rootedness and amoebas, derived for \(n = 1\) by Alon et al. (J Funct Anal 286(2):110226, 2024), to the case \(n > 1\) . The extension uses results in Bushueva and Tsikh (Proc Steklov Inst Math 270:52–63, 2012) about homology of complements of amoebas of algebraic sets of codimension \(> 1\) . Third, we prove that the divisors of all uniformly generic real-rooted P are FQ. The proof uses the formula relating Grothendieck residues and Newton polytopes derived by Gelfond and Khovanskii (Dokl Math 54(2):1–5, 1996). Finally, we note that Olevskii and Ulanovskii (Compt Rend Mat 358(11–12):1207–1211, 2020) have proved that all FQ with positive integer weights are divisors of real-rooted trigonometric polynomials for \(n = 1\) but that the situation for \(n > 1\) remains unsolved.
